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Ergodicity of PCA: Equivalence between Spatial and Temporal Mixing Conditions

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Ergodicity of PCA: Equivalence between Spatial and Temporal Mixing Conditions
2.	Creator	Author's name, affiliation, country	Pierre-Yves Louis; Technische Universitat Berlin
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)
4.	Description	Abstract	For a general attractive Probabilistic Cellular Automata on $S^{\mathbb{Z}^d}$, we prove that the (time-) convergence towards equilibrium of this Markovian parallel dynamics, exponentially fast in the uniform norm, is equivalent to a condition ($\mathcal{A}$). This condition means the exponential decay of the influence from the boundary for the invariant measures of the system restricted to finite boxes. For a class of reversible PCA dynamics on $\{-1;+1\}^{\mathbb{Z}^d}$ with a naturally associated Gibbsian potential $\varphi$, we prove that a (spatial-) weak mixing condition ($\mathcal{WM}$) for $\varphi$ implies the validity of the assumption ($\mathcal{A}$); thus exponential (time-) ergodicity of these dynamics towards the unique Gibbs measure associated to $\varphi$ holds. On some particular examples we state that exponential ergodicity holds as soon as there is no phase transition.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2004-10-07
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ecp.ejpecp.org/article/view/1116
10.	Identifier	Digital Object Identifier	10.1214/ECP.v9-1116
11.	Source	Journal/conference title; vol., no. (year)	Electronic Communications in Probability; Vol 9
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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